Let's solve the given multiplication problem using the FOIL method. The problem given is:
[tex]\[
(8\sqrt{2} + 4\sqrt{3})(7\sqrt{2} - 3\sqrt{3})
\][/tex]
We use the FOIL method, which stands for First, Outer, Inner, Last. This method helps in distributing each term in the first binomial to each term in the second binomial.
1. First terms:
[tex]\[
8\sqrt{2} \times 7\sqrt{2}
\][/tex]
2. Outer terms:
[tex]\[
8\sqrt{2} \times -3\sqrt{3}
\][/tex]
3. Inner terms:
[tex]\[
4\sqrt{3} \times 7\sqrt{2}
\][/tex]
4. Last terms:
[tex]\[
4\sqrt{3} \times -3\sqrt{3}
\][/tex]
The product of the first terms is:
[tex]\[
8\sqrt{2} \times 7\sqrt{2} = (8 \times 7) \times (\sqrt{2} \times \sqrt{2}) = 56 \times 2 = 112
\][/tex]
Thus, the product of the first terms is
[tex]\[
112
\][/tex]
